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- Warm Up
- 1. Find each angle measure.
- True or False. If false explain.
- 2. Every equilateral triangle is isosceles.
- 3. Every isosceles triangle is equilateral.

60 60 60

True

False an isosceles triangle can have only two

congruent sides.

Isosceles Triangles

- 4.2
- Apply properties of isosceles triangles.

Recall that an isosceles triangle has at least

two congruent sides. The congruent sides are

called the legs. The vertex angle is the angle

formed by the legs. The side opposite the vertex

angle is called the base, and the base angles are

the two angles that have the base as a side.

?3 is the vertex angle. ?1 and ?2 are the base

angles.

4-1

4-3

Theorem 4-2

- The bisector of the vertex of an isosceles

triangle is the perpendicular bisector of the

base

Example 1 Astronomy Application

The m?YZX 180 140, so m?YZX 40.

Since ?YZX ? ?X, ?XYZ is isosceles by the

Converse of the Isosceles Triangle Theorem.

Thus YZ YX 20 ft.

Example 2 Finding the Measure of an Angle

Find m?F.

m?F m?D x

Isosc. ? Thm.

m?F m?D m?A 180

? Sum Thm.

Substitute the given values.

x x 22 180

Simplify and subtract 22 from both sides.

2x 158

Divide both sides by 2.

x 79?

Thus m?F 79

Example 3 Finding the Measure of an Angle

Find m?G.

m?J m?G

Isosc. ? Thm.

Substitute the given values.

(x 44)? 3x?

Simplify x from both sides.

44 2x

Divide both sides by 2.

x 22?

Thus m?G 22 44 66.

Check It Out! Example 4

Find m?H.

m?H m?G x

Isosc. ? Thm.

m?H m?G m?F 180

? Sum Thm.

Substitute the given values.

x x 48 180

Simplify and subtract 48 from both sides.

2x 132

Divide both sides by 2.

x 66?

Thus m?H 66

Check It Out! Example 5

Find m?N.

m?P m?N

Isosc. ? Thm.

Substitute the given values.

(8y 16)? 6y?

Subtract 6y and add 16 to both sides.

2y 16

Divide both sides by 2.

y 8?

Thus m?N 6(8) 48.

The following corollary and its converse show the

connection between equilateral triangles and

equiangular triangles.

(No Transcript)

Example 6 Using Properties of Equilateral

Triangles

Find the value of x.

?LKM is equilateral.

Equilateral ? ? equiangular ?

The measure of each ? of an equiangular ? is 60.

(2x 32)? 60?

Subtract 32 both sides.

2x 28

Divide both sides by 2.

x 14

Example 7 Using Properties of Equilateral

Triangles

Find the value of y.

?NPO is equiangular.

Equiangular ? ? equilateral ?

Definition of equilateral ?.

5y 6 4y 12

Subtract 4y and add 6 to both sides.

y 18

Check It Out! Example 8

Find the value of JL.

?JKL is equiangular.

Equiangular ? ? equilateral ?

Definition of equilateral ?.

4t 8 2t 1

Subtract 4y and add 6 to both sides.

2t 9

t 4.5

Divide both sides by 2.

Thus JL 2(4.5) 1 10.

Lesson Quiz Part I

Find each angle measure. 1. m?R 2. m?P Find

each value. 3. x 4. y 5. x

28

124

6

20

26

Homework

- Pg. 191 1-8, 12-15, 18, 26-33

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